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Hopefully someone else can provide an easier answer or an app that automates this, but the method I've found so far only works for jailbroken phones:

Use iFile or some other file manager to access .amr files that are stored in /var/mobile/Library/Voicemail. The files are numbered sequentially and voicemail.db looks like its used to organize them.

Try to locate the correct .amr using timestamp.

Use file manager to email .amr file to myself or someone else.

A sidenote is that doing this I was able to find some orphaned .amr files that seemed to not have been completely deleted (.amr files are still stored when deleted until they are cleared from Deleted Messages). I was able to safely delete these old voicemails (over a year old, probably persisting from backups) from the folder.